Surrendering the a2-pawn isn't really necessary, but it's also a good choice.
11... a8c6  12. d1e1 Black failed to solve the king problem and is now strategically lost. White can play with c5, with d5, with h5, which are all interesting plans. Meanwhile, the black pieces lack harmony. 12... a5a3  13. h2h4  13... c6a4 